Advisable to buy Hakone Free Pass?Yes, the pass pays off if you do the round course around Hakone. But if you return from Gotemba directly to Tokyo, you should buy and use the pass from Odawara rather than from Tokyo because direct transportation between Gotemba and Tokyo is not covered by the pass.
Should I do the round trip on 1st day then to Gotemba on 2nd day?
Yes, that sounds like a good plan.
How to go to Gotemba from Hakone Yumoto?
It is explained on the following page:
http://www.japan-guide.com/e/e5213.html
Is the fare cover by Hakone Free pass?
Yes.
Can I travel back to Shinjuku from Gotemba?
Yes, but it is not covered by the Hakone Free Pass, as explained above.
It is a big round to go back to Hakone Yumoto then to Shinjuku.
Yes, that would be quite a big detour.
I would like to visit Kamakura. Should I do Gotemba - Kamakura on 2nd day or make another day trip from Shinjuku?
It depends on your preferred pace of travel. If you prefer a medium to slow pace, I definitely recommend to visit the two places on separate days, but if you like to spend only a little bit of time in each place, the two could theoretically be packed into a single day.
